Redirected Walking (RDW) enables continuous locomotion in virtual environments (VEs) within limited physical spaces. However, classic RDW methods rely on static settings with tight boundaries, which can reduce their applicability in large shared physical spaces where boundary constraints are not dominant, especially under dynamic and multi-user conditions. To overcome this, we introduce a new task: Redirected Walking in Open Physical Spaces (OPSRDW), which allows users to navigate expansive VEs safely and naturally despite dynamic obstacles and without requiring a fixed boundary model. We further propose Dynamic Control and Redirection with Safety Constraints (DyCoRe), which formulates OPS-RDW as a constrained optimization problem. DyCoRe uses Dynamic Control Barrier Functions to model real-time collision avoidance constraints and solves an online Quadratic Programming problem to compute optimal velocities that minimize path deviation while ensuring safety. These velocities are mapped into real-time redirection gains, guiding users along natural paths while reducing collision risk. A relaxation mechanism is incorporated to handle infeasible scenarios. Extensive simulations suggest consistent improvements in safety and obstacle clearance over state-of-theart methods. User studies demonstrate that DyCoRe significantly improves navigation continuity, reduces the number of resets, and tends to reduce perceived discomfort compared to classic RDW strategies. DyCoRe provides an efficient and learning-free solution for safe and natural VR locomotion in open physical spaces with dynamic obstacles and multiple users.

Symmetric positive definite(SPD) matrices have shown important value and applications in statistics and machine learning, such as FMRI analysis and traffic prediction. Previous works on SPD matrices mostly focus on discriminative models, where predictions are made directly on E(X|y), where y is a vector and X is an SPD matrix. However, these methods are challenging to handle for large-scale data. In this paper, inspired by denoising diffusion probabilistic model(DDPM), we propose a novel generative model, termed SPD-DDPM, by introducing Gaussian distribution in the SPD space to estimate E(X|y). Moreover, our model can estimate p(X) unconditionally and flexibly without giving y. On the one hand, the model conditionally learns p(X|y) and utilizes the mean of samples to obtain E(X|y) as a prediction. On the other hand, the model unconditionally learns the probability distribution of the data p(X) and generates samples that conform to this distribution. Furthermore, we propose a new SPD net which is much deeper than the previous networks and allows for the inclusion of conditional factors. Experiment results on toy data and real taxi data demonstrate that our models effectively fit the data distribution both unconditionally and conditionally.